The present invention is generally related to analysis of structural integrity of structures via computer implemented simulation, and more particularly to analysis and design of integrity of structures via computer implemented simulation by prescribing modified second-order tetrahedral elements during small and finite deformation.
The last two decades have seen major changes in the application of the finite element method. During the seventies the range of elements and their performance was improved. Application of the method was still fairly new and the learning curve was steep. Much of the change, particularly in the functionality of programs, was influenced by response from the growing number of users. The advances in the eighties were greatly influenced by the increased power and reduced cost of computers, and the development of computer graphics.
Development of solution techniques slowed, but powerful graphics-based techniques for the modeling of complex structures were developed. Computer power hungry techniques for the solution of nonlinear problems became a practical reality. Development in the nineties has been dominated by the automation of the modeling process, involving adaptive mesh refinement and design optimization. These have led to a return to the theory to establish error estimating techniques.
The Finite Element (FE) method has been used with great success to solve many types of problems including:
stress analysis
dynamic response
heat transfer
In recent years there has been an increasing requirement for structures and safety related equipment to be seismically qualified. Smaller items can be tested on shake tables on which they are subjected to dynamic loading, simulating the effect of an earthquake. Large structures cannot undergo this type of test and it is necessary to simulate the behavior analytically, often using the FE technique.
Destructive testing and service failures often provide very limited information. They may show the weakest part of a structure but give no clear indication of the most effective remedial action, where another failure might occur after redesign, or where material could be removed without detriment.
A similar approach was adopted in the analysis of certain types of structures, such as building frames, process piping and aircraft structures. The structure was broken down into smaller parts or elements for which exact results were available, and the behavior of the structure as a whole found by the solution of a set of simultaneous equations.
Although originally intended to represent sections of structures which were in reality quite discrete, the early elements were used with considerable success to analyze more general structures. Modern, general-purpose finite element programs have libraries of elements which permit many different geometries to be modeled. They can be broken down into four main categories:
line elements, such as beams, struts and pipes
two-dimensional solid elements for modeling plane strain and axisymmetry
three-dimensional solid elements for modeling any solid shape
shell elements suitable for modeling thin structures
Common finite elements are illustrated in FIGS. 1A-1D, and examples of their use in structures are shown in FIGS. 2A-2D.
Three-dimensional solid elements are probably the simplest to understand. As in the earlier methods described above, the structure is represented by an assemblage of notional elements, which, for the sake of visualization, may be likened to building blocks.
Unlike an element that is exact, the stiffness of solid elements are calculated approximately by numerical integration, based on assumptions about how the element deforms under loads at the node points. Providing that the elements are sufficiently small, the error due to these approximations is acceptable.
For design by analysis it is necessary to postulate sizes and thicknesses first, and then calculate the stresses. The process is frequently iterative, with changes of size or thicknesses in each loop until a satisfactory or optimized design is achieved. The finite element method lends itself readily to this process since it is often possible to change just a single or few numerical values in the input data to change sizes or thicknesses.
One computer simulation process used for structural design is the use of tetrahedral elements in the finite element analysis and simulation of structures. This is particularly desirable because automatic mesh-generation techniques are now available to subdivide general objects of any shape into meshes of tetrahedral elements. Second-order tetrahedral elements usually give accurate results in small and finite deformation problems with no contact. However, we have discovered that these elements are not appropriate for contact problems because in uniform pressure situations the contact forces are non-uniform at the corner and midside nodes.
We have also discovered that, for second-order tetrahedra, the contact forces at the corner nodes are zero, while the midside nodes carry all the contact load. The zero contact forces at the corner nodes in the tetrahedra result in zero contact pressures. As an additional complication, we have determined that non-convergence of contact conditions may result with second-order tetrahedral elements. We have also discovered that the second-order tetrahedra elements may exhibit significant volumetric locking when incompressibility is approached.
In comparison, the first-order tetrahedra produce uniform contact forces and pressures, but overall results can be very inaccurate due to severe volumetric and shear locking. In addition, very fine meshes may be needed for first-order tetrahedra elements to attain results of sufficient accuracy.
We have also realized that it is desirable to simulate and analyze the structural integrity of structures via computer implemented simulation by prescribing modified second-order tetrahedral elements during small and finite deformation.
We have also realized that it is desirable to simulate and analyze the structural integrity of structures by prescribing modified second-order tetrahedral elements that provide robustness and convergence for contact simulations.
We have also realized that it is desirable to utilize second-order tetrahedra elements that minimize volumetric locking when incompressibility is approached.
It is a feature and advantage of the present invention to simulate and analyze structural integrity of structures via computer implemented simulation.
It is another feature and advantage of the present invention to simulate and analyze the structural integrity of structures via computer implemented simulation by prescribing modified second-order tetrahedral elements during small and finite deformation.
It is another feature and advantage of the present invention to simulate and analyze the structural integrity of structures by prescribing modified second-order tetrahedral elements that provide robustness and convergence for contact simulations.
It is another feature and advantage of the present invention to utilize second-order tetrahedra elements that minimize volumetric locking when incompressibility is approached.
The present invention utilizes finite element analysis to provide the additional simulation benefits of:
optimization of product performance
optimization of product cost
reduction of development time
elimination of reduction of testing
achievement of required quality first time
improved safety
satisfaction of design codes
improved information for engineering decisions
fuller understanding of components allowing more rational design
The present invention is based, in part, on our discovery of a formulation of a modified second-order tetrahedral element which addresses the shortcomings mentioned above. The element gives rise to uniform contact forces and pressures, exhibits minimal shear and volumetric locking, and is robust during finite deformation. A satisfactory xe2x80x9clumpedxe2x80x9d mass matrix can also be generated for dynamic applications. In addition, a hybrid version of this element is provided for use with incompressible and nearly incompressible constitutive models.
The present invention is also based, in part, on our discovery that it is possible to use a combination of hexahedral elements, while the user only sees or deals with a second-order tetrahedral element. Advantageously, we have divided or broken the standard tetrahedral object into four hexahedral objects. The user sees the second-order tetrahedral element with four vertices, and six mid-edge points.
The computer simulation advantageously constrains the four points that are located in the mid-faces of the tetrahedral elements to the three vertices and three mid-edge points associated with the faces of the tetrahedral element.
The computer simulation advantageously indirectly constrains the midbody point using an hourglass technique, instead of placing direct constraints thereon. The use of the hourglass control greatly relieves volumetric and shear locking, thereby facilitating convergence, resulting in a robust family of modified second-order tetrahedral elements. The present invention thus provides simulation of a second-order tetrahedra using a combination of hexahedral finite elements with constraints on the four mid-face points and the mid-body point.
In accordance with one embodiment of the invention, a computer implemented process prescribes second-order tetrahedral elements during simulation in the design analysis of structure. The computer implemented process includes the steps of defining a finite element model for an object including at least one tetrahedral element, and defining the at least one tetrahedral element as a combination of hexahedral sub-elements. The computer implemented process also includes the steps of executing the simulation, and evaluating the structure for structural integrity responsive thereto.
In accordance with another embodiment, a computer readable tangible medium stores instructions for implementing a process driven by a computer. The instructions control the computer to perform the process of prescribing second-order tetrahedral elements during small or finite deformation simulation in the design analysis of structures, as described in detail below.